SOLUTION: 4) CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pu

Question 69027: 4) CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Crane came out and gratefully thanked the traveling salesman for saving his daughter�s life. Mr. Crane insisted on giving the man an award for his heroism.
So, the salesman said, �If you insist, I do not want much. Get your checkerboard and place one grain of wheat on the first square. Then place two grains of wheat on the next square. Then place four grains on the third square. Continue this until all 64 squares are covered with grains of wheat.� As he had just harvested his wheat, Mr. Crane did not consider this much of an award, but he soon realized he made a miscalculation on the amount of wheat involved.
a) How much wheat would Mr. Crane have to put on the 24th square?
Answer:
Show work in this space.

b) How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?
Answer:
Show work in this space.

c) Calculate the amount of wheat necessary to fill the whole checkerboard (64 squares). How much wheat would the farmer need to give the salesman? Please provide the answer in either scientific notation, or calculate and show all 20 digits.
Answer:

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If you look at the relationship between the number of wheat grains on the squares you'll see that the number is growing exponentially. Going from 1 to 2, the number is doubled. Continuing from 2 to 4, again the number is doubled. So to get to any term you must double the previous value. So this problem involves powers and exponents. If you wanted to get to the 10th term, you would start at the first term and double each term to get to the tenth term. To get to the 10th term, you must multiply 2 by itself 10 times. So to get to the 64th term, you must multiply 2 by itself 64 times, see the pattern? To get the pattern down officially, the sequence is where n is any term you pick. We start off at n=0 to get the first term of 1 and we move from there. Hope a little background on this helps, so here we go.
a)To find out how many grains of wheat are on the 24th square, simply evaluate 2^23 (we go to the n-1 term since we started off at n=0). Well it comes out to 8,388,608 grains of wheat.
If you want to verify, you can double 2 to 4, 4 to 8, etc until you get there.
b)This one might be a little tricky, but I'll give it a shot. You could painstakingly add up 2,4,8...8,388,608 but that's probably not going to happen.
Also, I don't know if its just me, but this seems a little advanced for algebra. Now I could go into great detail of how to derive this formula, but I'm not. To find the sum of any geometric series, i.e. how to find 1+2+4+8+...8,388,608, there is a neat shortcut. If you're curious search "Sum of geometric series". Anyway, if I have a series of n terms the sum is
don't worry about a, we will ignore it, a=1 right now
and r is the factor to go from term to term, in this case r=2 (we're doubling everything).
So

so this means that there are a total of 16,777,215 grains of wheat if 24 squares were filled.
c)To find how many grains would fill the entire board, use the same formula but with 64 squares

So there are 18,446,744,073,709,551,616 grains of wheat on the board
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